We shall now sketch the proof of Theorem REF  emphasizing the main steps and differences compared with [1]}.
In order to flip lattices 2 and 3, we conjugate the Hamiltonian by \(P = \operatorname{diag}(1,\sigma _1,1)\)  such that \(H_{w_0}(\vartheta ) := PH_{w_0}(\theta ,\phi =\tfrac{1}{4},\vartheta )P\)  becomes
\(H_{w_0}(\vartheta ) = \begin{pmatrix} 0 & w_0U(\tfrac{x-\phi }{L}) & K(\theta )& 0 \\w_0U(\tfrac{x-\phi }{L}) &0 &0 &K(\theta ) \\K(\theta )^* &0&0 & w_0U(\tfrac{x+\phi }{L}) \\0&K(\theta )^* &w_0U(\tfrac{x+\phi }{L}) & 0 \end{pmatrix}.\) 
